Sketch an equilateral triangle and plot a point at the center of it.

Note that if you rotate the triangle 120 degrees about this point the triangle looks the same as it initially did. This is the symmetry. Similarly for . ( would be the "identity" transformation. Obviously if you rotate the triangle by 0 degrees it looks the same as it did before you "rotated" it.)

For the "flips" sketch a line from the center of the triangle through vertex A. Note that we have reflection symmetry over this line: if we reflect the triangle over this line it looks the same as it did before. This is the element . Obviously we have the same kind of symmetry no matter which vertex we pick, so we also have and . (The and are symmetries about "horizontal" and "vertical" lines. These symmetries depend on the orientation of the square, obviously.)

See what you can do with these rotations and reflections for a pentagon. (Hint: You'll have 5 rotations and 5 reflections.)

-Dan